A probability density function (PDF) generalizes the discrete probability mass function to continuous random variables, representing probability per unit length rather than probability itself, such that probabilities are obtained by integrating the density over an interval rather than summing point masses; it must be non-negative and integrate to one over its support, and it relates to the cumulative distribution function (CDF) as its derivative (equivalently, the CDF is the antiderivative of the PDF via the Fundamental Theorem of Calculus). The uniform distribution, the simplest continuous distribution with constant density on a bounded interval, illustrates these definitions and motivates the Law of the Unconscious Statistician (LOTUS) for computing expectations of transformed random variables without deriving their distributions, as well as the "universality of the uniform," a theorem showing that any distribution can be generated by applying the inverse CDF to a uniform random variable. This belongs to probability theory, specifically the transition from discrete to continuous distribution theory.
A probability density function (PDF) generalizes the discrete probability mass function to continuous random variables, representing probability per unit length rather than probability itself, such t…